In terms of robustness for the Dirac degree threshold, for graphs a result of Sudakov and Vu [1]} shows that for \(p=\omega (\log n/n)\)  any subgraph of \(\mathbb {G}(n,p)\)  with minimum degree \((1/2+o(1))np\)  has a perfect matching. This local resilience property is strictly stronger than the robustness of thm:pm in the case \((k,\ell )=(2,1)\)  since such a “minimum-degree subgraph” can be specified by intersecting the minimum degree host with the random graph \(\mathbb {G}(n,p)\) . It remains a problem of substantial interest to derive any such resilient threshold (or closely related universality result) from spread related techniques.
